论文标题
Lebesgue空间的矢量平衡
Vector Balancing in Lebesgue Spaces
论文作者
论文摘要
离散数学中的诱人猜想是Komlós之一，这表明对于任何向量$ \ MathBf {a} _1，\ ldots，\ Mathbf {a} _n \ in B_2^m $ in $美元问$ \ ell_q $ -norm绑定到$ \ mathbf {a} _1，\ ldots，\ mathbf {a} _n \ in B_p^m $是什么，这是自然的扩展。我们证明，以$ 2 \ le p \ le q \ le \ le \ infty $，此类向量承认分数着色$ x_1，\ dots，x_n \ in [-1,1] $，线性数为$ \ pm 1 $ coordinate \ leq o（\ sqrt {\ min（p，\ log（2m/n））}）\ cdot n^{1/2-1/p + 1/q} $，并且可以以$ \ frac {1/2-1/p + 1/p + 1/q} $的另一个因素而获得完整的着色。特别是，对于$ p \ in（2,3] $，我们确实可以找到符号$ \ mathbf {x} \ in \ {-1,1,1 \}^n $带有$ \ | \ | \ sum_ {i = 1}^n x_i \ x_i \ mathbf {a} \ frac {1} {p-2}）$。 此外，我们证明，对于任何固定常数$δ> 0 $，在中央对称的主体中，$ k \ subseteq \ mathbb {r}^n $，至少$ e^{ - δn} $一个人可以在多项典型的时间中找到这样的分数着色。以前，这仅是一个足够小的常数而闻名的 - 实际上，在这种制度中，经典的非构造性参数不应用，并且形式的$ \ mathbf {x} \ in \ in \ {-1,0,1,1 \}^n $不一定存在。
A tantalizing conjecture in discrete mathematics is the one of Komlós, suggesting that for any vectors $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_2^m$ there exist signs $x_1, \dots, x_n \in \{ -1,1\}$ so that $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_\infty \le O(1)$. It is a natural extension to ask what $\ell_q$-norm bound to expect for $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_p^m$. We prove that, for $2 \le p \le q \le \infty$, such vectors admit fractional colorings $x_1, \dots, x_n \in [-1,1]$ with a linear number of $\pm 1$ coordinates so that $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_q \leq O(\sqrt{\min(p,\log(2m/n))}) \cdot n^{1/2-1/p+ 1/q}$, and that one can obtain a full coloring at the expense of another factor of $\frac{1}{1/2 - 1/p + 1/q}$. In particular, for $p \in (2,3]$ we can indeed find signs $\mathbf{x} \in \{ -1,1\}^n$ with $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_\infty \le O(n^{1/2-1/p} \cdot \frac{1}{p-2})$. Our result generalizes Spencer's theorem, for which $p = q = \infty$, and is tight for $m = n$. Additionally, we prove that for any fixed constant $δ>0$, in a centrally symmetric body $K \subseteq \mathbb{R}^n$ with measure at least $e^{-δn}$ one can find such a fractional coloring in polynomial time. Previously this was known only for a small enough constant -- indeed in this regime classical nonconstructive arguments do not apply and partial colorings of the form $\mathbf{x} \in \{ -1,0,1\}^n$ do not necessarily exist.
